Topological colouring of fluid particles unravels finite-time coherent sets

Charó, Gisela D.; Artana, Guillermo; Sciamarella, Denisse

doi:10.1017/jfm.2021.561

Cited by 8 publications

(15 citation statements)

References 70 publications

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“…We left untouched this last rank, mostly because this is still an open problem for most of toroidal chaos, in spite of the recent success for two simple cases [Mangiarotti & Letellier, 2021]. We left for future works the promising results obtained with the Betti numbers and their extension [Charó et al, 2020[Charó et al, , 2021.…”

Section: Discussionmentioning

confidence: 99%

Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

Letellier,

Stankevich,

Rössler

2021

Preprint

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Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine what are the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labelling. Addressing these problems correspond to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three-and four-dimensional spaces, covering a large variety of known (and less known) examples of chaotic systems. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map...). By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold -the highest taxonomic rank for classifying chaotic attractor -by a linking matrix (or linker) to multi-component attractors (bounded by a torus whose genus g ≥ 3).

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Section: Discussionmentioning

confidence: 99%

Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

Letellier,

Stankevich,

Rössler

2021

Preprint

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“…Such flow separators are associated with LCSs that are known to separate dynamically distinct regions in fluid flows (Kelley et al, 2013). authors have also successfully used BraMAH to study numerically generated fluid particle behavior in the wake behind a rotary oscillating cylinder (Charó et al, 2021a).…”

Section: )mentioning

confidence: 99%

Review Article: Dynamical Systems, Algebraic Topology, and the Climate Sciences

Ghil

Sciamarella

2023

Preprint

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Abstract. The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports; and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefitted from remarkable technological advances in computing resources, in throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of nonautonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change; and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing.

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“…The important message here is that a finer description can be obtained when one is working with a uniformly oriented complex. 36 Table I Branched manifolds are mathematical objects which are not Hausdorff. In order to endow the corresponding complex with a uniform orientation that is compatible with the flow, one must decompose the complex in sub-units, as a template must be decomposed into strips.…”

Section: B Homology Groupsmentioning

confidence: 99%

“…Here, we use a non-simplicial algebraic code to compute homologies, as in most of our works. [33][34][35][36]62 As all complexes formed from clouds of points, a BraMAH complex will depend on the number of points in P, that is, in this work, on the integration time window and time step. Tailored so that the nature of its highest dimensional cells is tied to the local dimension and curvature of the underlying branched manifold, a BraMAH complex is topologically faithful to the latter.…”

Section: Supplementary Materialsmentioning

confidence: 99%

“…The extended algorithm identifies the k-generators of the homology groups and introduces orientability chains. 33,34 More recent applications of this approach, now called BraMAH (Branched Manifold Analysis through Homologies), incorporate the extraction of weak boundaries, 35,36 enabling a more precise description of four-dimensional manifolds. Note also that Conley index was used for constructing a symbolic dynamics for thick first-return map.…”

Section: Introductionmentioning

confidence: 99%

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Templex: A bridge between homologies and templates for chaotic attractors

Charó

Letellier

Sciamarella

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest-dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated—namely, the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor, and a four-dimensional system. A link is established with their description in terms of templates.

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Topological colouring of fluid particles unravels finite-time coherent sets

Abstract: Abstract

Cited by 8 publications

References 70 publications

Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

Review Article: Dynamical Systems, Algebraic Topology, and the Climate Sciences

Templex: A bridge between homologies and templates for chaotic attractors

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